On the integral kernels of derivatives of the Ornstein–Uhlenbeck semigroup

Journal article (2016)

Authors

J.J.B. Teuwen Analysis -

Research Group

Analysis () (TU Delft)

Hermite polynomials Ornstein–Uhlenbeck Mehler kernel Gaussian measure Caldéron reproducing formula

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Published Date

2016

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

Abstract

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup e tL etL. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers L N LN of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for L L. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.